Final answer:
To determine if a function is a solution to a differential equation, we substitute the function into the equation and check for equality. By doing this for the given function and differential equation, we find that the function satisfies the equation.
Step-by-step explanation:
To determine whether the given function is a solution to the given differential equation, we need to substitute the function into the differential equation and check if both sides are equal. The given function is y = sinx + x^5, and the differential equation is (d^2y/dx^2) + y = x^5 + 20x^3.
First, we find the second derivative of y: d^2y/dx^2 = 5x^4 + cosx.
Substituting y and its second derivative into the differential equation, we get (5x^4 + cosx) + (sinx + x^5) = x^5 + 20x^3.
Simplifying, we have 5x^4 + cosx + sinx + x^5 = x^5 + 20x^3.
Canceling out x^5 from both sides, we are left with 5x^4 + cosx + sinx = 20x^3.
Therefore, the given function y = sinx + x^5 satisfies the differential equation.